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I'm currently working on a mini personal project on representation theory of $S_n$ over positive characteristic. I'm only an amateur (am a violin maker by profession) so please be nice to me!

Suppose $A$ is a semisimple algebra over a field $\mathbb{F}$, and let $\rho_1: A\rightarrow Mat_n(\mathbb{F})$ and $\rho_2:A\rightarrow Mat_n(\mathbb{F})$ be irreducible representations with $V_1$ and $V_2$ the corresponding left $A$ modules. Is it necessarily true that $tr(\rho_1(x))=tr(\rho_2(x))$ for all $x\in A$ implies that $V_1$ and $V_2$ are isomorphic $A$ modules?

Emil Sinclair
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  • Hi: this will be the second time I've left a comment for you mentioning that bare problem statements are not ordinarily acceptable. Please take a look at the advice we have for enriching a question with context. If you continue with this pattern, you will probably experience more pushback, and less patience. Thanks. – rschwieb Jul 11 '19 at 13:24
  • The first sentence you added about your project is helpful, but the second sentence you added is not helpful. What about your thoughts on the problem? That would be FAR more helpful. – rschwieb Jul 11 '19 at 13:45
  • Really sorry to disappoint you. Obviously I will be the disappointed one if nobody answers me, but I come up with random questions in my head and post them if I can't answer them. Most of the time there's no real context. – Emil Sinclair Jul 11 '19 at 13:50
  • The problem is that in the past, people have "just come up with random questions" that look particularly like homework they're looking to fob off on others, so the community gradually grew wary of this. In fact, I think this is an interesting question because I have read that group characters determine representation isomorphy exactly in characteristic $0$, and I'm wondering if the same holds for algebra representations. – rschwieb Jul 11 '19 at 13:57
  • You wrote $Mat_n(\mathbb F)$ for both representations : is it assumed that the two representations have the same dimension ? (note that in positive characteristic you cannot determine the dimension from the character by looking at $1$, and this is a way to answer the question negatively if you don't a priori assume the dimensions to be equal) – Maxime Ramzi Jul 11 '19 at 17:19
  • Yes, it is. I also assumed that $V_1$ and $V_2$ are simple modules, so I'm not allowing things like adding $p$ times the same representation. – Emil Sinclair Jul 11 '19 at 18:22

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No, not necessarily. For instance, let $K$ be a finite inseparable field extension of $\mathbb{F}$ and let $A=K\times K$. Then there are two non-isomorphic simple $A$-modules $V_1,V_2$ given by the projections onto the two coordinates. However, the characters of both these representations are identically $0$: the character of $V_i$ is just the $i$th projection $A\to K$ followed by the field trace $K\to \mathbb{F}$, but the field trace is $0$ since $K$ is inseparable over $\mathbb{F}$.

Eric Wofsey
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